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Counting Nodes in Smolyak Grids

Jocelyn Minini, Micha Wasem

TL;DR

The paper addresses counting nodes in Smolyak grids generated from univariate rules. It develops a generating-function framework and uses dimension-wise induction to derive closed-form expressions for node counts across several growth functions, notably f(k)=n^k-1, f(k)=n^k, f(k)=n^k+1, and f(k)=k. The main contributions are the explicit formulas for Nd(mu) and Nbar_d(mu) and their generalizations of prior results by Novak, Müller-Gronbach, Bungartz-Griebel, and Ullrich, enabling exact cost estimates for Smolyak grids. This framework supports both nested and non-nested sequences and has practical implications for efficient implementation and planning in high-dimensional quadrature and interpolation tasks.

Abstract

Using generating functions, we are proposing a unified approach to produce explicit formulas, which count the number of nodes in Smolyak grids based on various univariate quadrature or interpolation rules. Our approach yields, for instance, a new formula for the cardinality of a Smolyak grid, which is based on Chebyshev nodes of the first kind and it allows to recover certain counting-formulas previously found by Bungartz-Griebel, Kaarnioja, Müller-Gronbach, Novak-Ritter and Ullrich.

Counting Nodes in Smolyak Grids

TL;DR

The paper addresses counting nodes in Smolyak grids generated from univariate rules. It develops a generating-function framework and uses dimension-wise induction to derive closed-form expressions for node counts across several growth functions, notably f(k)=n^k-1, f(k)=n^k, f(k)=n^k+1, and f(k)=k. The main contributions are the explicit formulas for Nd(mu) and Nbar_d(mu) and their generalizations of prior results by Novak, Müller-Gronbach, Bungartz-Griebel, and Ullrich, enabling exact cost estimates for Smolyak grids. This framework supports both nested and non-nested sequences and has practical implications for efficient implementation and planning in high-dimensional quadrature and interpolation tasks.

Abstract

Using generating functions, we are proposing a unified approach to produce explicit formulas, which count the number of nodes in Smolyak grids based on various univariate quadrature or interpolation rules. Our approach yields, for instance, a new formula for the cardinality of a Smolyak grid, which is based on Chebyshev nodes of the first kind and it allows to recover certain counting-formulas previously found by Bungartz-Griebel, Kaarnioja, Müller-Gronbach, Novak-Ritter and Ullrich.
Paper Structure (7 sections, 3 theorems, 40 equations, 1 figure)

This paper contains 7 sections, 3 theorems, 40 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Counting Nodes
  3. Growth Function $f(k) = n^{k}-1$
  4. Growth Function $f(k) = n^k$
  5. Growth Function $f(k) = n^{k}+1$
  6. Growth Function $f(k) = k$
  7. Further Remarks

Key Result

Theorem 1.1

It holds that $G_d(x) = G_1(x)^d(1-x)^{d-1} \text{ and }\mathcal{G}_d(x) = \mathcal{G}_1(x)^d,$ where

Figures (1)

  • Figure 1: Smolyak grids with cardinality for $d=2$: (a) $9$, (b) $45$ (c) $189$ and in $d=3$: (d) $27$, (e) $189$, (f) $999$ respectively.

Theorems & Definitions (10)

  • Theorem 1.1
  • Example
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark
  • Remark
  • Remark
  • Remark